Input PDF parametrisation and priors

An important part of PDF fitting is defining a useful parametrisation for the PDF shapes, as well as meaningful prior distributions that encode our knowledge of the problem.

In this notebook, we explore two different approaches:

• Full Dirichlet
• Valence shape + Dirichlet

In the end, it seems like the latter option makes more sense for us and is therefore implemented elsewhere in the PartonDensity package. We demonstrate why below.

using Distributions, Plots, SpecialFunctions, Printf
const sf = SpecialFunctions;

"Full Dirichlet" approach

A clean way to ensure the momentum sum rule would be to sample different contributions of the momentrum density integral from a Dirichlet distribution, then use these weights to set the parameters on the individual Beta distributions. However, in practice this is non-trvial as we also want to fix the normalisation of the number densities of the valance contributions.

9 components of decreasing importance

dirichlet = Dirichlet([3., 2., 1, 0.5, 0.3, 0.2, 0.1, 0.1, 0.1])
data = rand(dirichlet, 1000);

Have a look

plot()
for i in 1:9
    histogram!(data[i,:], bins=range(0, stop=1, length=20), alpha=0.7)
end
plot!(xlabel="I_i = A_i B_i")

This would be great as the sum rule is automatically conserved

sum(data, dims=1)

1×1000 Matrix{Float64}:
 1.0  1.0  1.0  1.0  1.0  1.0  1.0  1.0  …  1.0  1.0  1.0  1.0  1.0  1.0  1.0

But, it is non-trival to define valence params from this

I = rand(dirichlet)

9-element Vector{Float64}:
 0.6184963926254953
 0.22901665127898943
 0.0035830406685615317
 0.005179999555751733
 0.0008791995708309679
 2.0213277714499948e-8
 0.1428446879331321
 4.38059442234342e-15
 8.153956912606594e-9

Valance u component

λ_u = rand(Uniform(0, 1))
K_u = rand(Uniform(0, 10))

6.644263843664793

Integral of number density must = 2

A_u = 2 / sf.beta(λ_u, K_u+1)

1.5762731652042588

integral of momentum density can be fixed by I[1]

I_1 = A_u * sf.beta(λ_u+1, K_u+1);

Could use a root-finder to find Ku given I1 and λ_u... Could be nasty to sample from though...

I_1 = 2 * (sf.beta(λ_u+1, K_u+1)/sf.beta(λ_u, K_u+1));

using Roots

function func_to_solve(K_u)
    return I_1 - 2 * (sf.beta(λ_u+1, K_u+1) / sf.beta(λ_u, K_u+1))
end

K_u ≈ find_zero(func_to_solve, (0, 10), Bisection())

true

While this approach might be nice, there are two issues in practice:

• It is difficult to set sensible priors on $\lambda_u$ that imply priors on $K_u$, and similarly for $\lambda_d$ and $K_d$
• The problem is overconstrained and we hav to use a root finder. This is rather fragile, and could fail for certain parameter combinations, such as we might find in a fit.

"Valence shape + Dirichlet" approach

We can handle this more elegantly (maybe?) by specifying constraints on the valence params through the shape of their Beta distributions, then using a Dirichlet to specify the weights of the gluon and sea components. The problem here is it isn't clear how to specify that the d contribution must be less than the u contribution, but it is possible to do this indirectly through priors on the shape parameters. This will however require some further investigation.

x = range(0, stop=1, length=50)

0.0:0.02040816326530612:1.0

High-level priors Looks like we maybe want to change lambda and K priors to boost these components

λ_u = 0.7 #rand(Uniform(0, 1))
K_u = 4 #rand(Uniform(2, 10))
λ_d = 0.5 #rand(Uniform(0, 1))
K_d = 6 #rand(Uniform(2, 10))

u_V = Beta(λ_u, K_u+1)
A_u = 2 / sf.beta(λ_u, K_u+1)

d_V = Beta(λ_d, K_d+1)
A_d = 1 / sf.beta(λ_d, K_d+1)

1.46630859375

Integral contributions

I_u = A_u * sf.beta(λ_u+1, K_u+1)
I_d = A_d * sf.beta(λ_d+1, K_d+1)

plot(x, x .* A_u .* x.^λ_u .* (1 .- x).^K_u * 2, alpha=0.7, label="x u(x)", lw=3)
plot!(x, x .* A_d .* x.^λ_d .* (1 .- x).^K_d, alpha=0.7, label="x d(x)", lw=3)
plot!(xlabel="x", legend=:topright)

@printf("I_u = %.2f\n", I_u)
@printf("I_d = %.2f\n", I_d)

I_u = 0.25
I_d = 0.07

The remaining 7 integrals can be dirichlet-sampled with decreasing importance

remaining = 1 - (I_u + I_d)
dirichlet = Dirichlet([3., 2., 1, 0.5, 0.3, 0.2, 0.1])
I = rand(dirichlet) * remaining;
sum(I) ≈ remaining

true

Gluon contributions

λ_g1 = rand(Uniform(-1, 0))
λ_g2 = rand(Uniform(0, 1))
K_g = rand(Uniform(2, 10))
A_g2 = I[1] / sf.beta(λ_g2+1, K_g+1)
A_g1 = I[2] / sf.beta(λ_g1+1, 5+1);

Sea quark contributions

λ_q = rand(Uniform(-1, 0))
A_ubar = I[3] / (2 * sf.beta(λ_q+1, 5+1))
A_dbar = I[4] / (2 * sf.beta(λ_q+1, 5+1))
A_s = I[5] / (2 * sf.beta(λ_q+1, 5+1))
A_c = I[6] / (2 * sf.beta(λ_q+1, 5+1))
A_b = I[7] / (2 * sf.beta(λ_q+1, 5+1));

total = A_u * sf.beta(λ_u+1, K_u+1) + A_d * sf.beta(λ_d+1, K_d+1)
total += A_g1 * sf.beta(λ_g1+1, 5+1) + A_g2 * sf.beta(λ_g2+1, K_g+1)
total += 2 * (A_ubar + A_dbar + A_s + A_c + A_b) * sf.beta(λ_q+1, 5+1)
total ≈ 1

true

x = 10 .^ range(-2, stop=0, length=500)

500-element Vector{Float64}:
 0.01
 0.010092715146305713
 0.010186289902446875
 0.010280732238308653
 0.010376050197669118
 0.010472251898884349
 0.010569345535579883
 0.010667339377348576
 0.010766241770454934
 0.010866061138545973
 ⋮
 0.92882922501725
 0.9374408787662996
 0.946132375589077
 0.9549044557518083
 0.963757866384109
 0.9726933615426174
 0.9817117022752193
 0.9908136566858671
 1.0

How does it look?

xg2 = A_g2 * x.^λ_g2 .* (1 .- x).^K_g
xg1 = A_g1 * x.^λ_g1 .* (1 .-x).^5
plot(x, x .* A_u .* x.^λ_u .* (1 .- x).^K_u * 2, alpha=0.7, label="x u(x)", lw=3)
plot!(x, x .* A_d .* x.^λ_d .* (1 .- x).^K_d, alpha=0.7, label="x d(x)", lw=3)
plot!(x, xg1 + xg2, alpha=0.7, label="x g(x)", lw=3)
plot!(x, A_ubar * x.^λ_q .* (1.0 .- x).^5, alpha=0.7, label="x ubar(x)", lw=3)
plot!(x, A_dbar * x.^λ_q .* (1.0 .- x).^5, alpha=0.7, label="x dbar(x)", lw=3)
plot!(x, A_s * x.^λ_q .* (1.0 .- x).^5, alpha=0.7, label="x s(x)", lw=3)
plot!(x, A_c * x.^λ_q .* (1.0 .- x).^5, alpha=0.7, label="x c(x)", lw=3)
plot!(x, A_b * x.^λ_q .* (1.0 .-  x).^5, alpha=0.7, label="x b(x)", lw=3)
plot!(xlabel="x", legend=:bottomleft, xscale=:log, ylims=(1e-8, 10), yscale=:log)

Prior predictive checks

We can start to visualise the type of PDFs that are allowed by the combination of the choice of parametrisation and prior distributions with some simple prior predictive checks, as done below...

N = 100
alpha = 0.03
total = Array{Float64, 1}(undef, N)
first = true
leg = 0

plot()
for i in 1:N

    λ_u_i = rand(Uniform(0, 1))
    K_u_i = rand(Uniform(2, 10))
    λ_d_i = rand(Uniform(0, 1))
    K_d_i = rand(Uniform(2, 10))
    A_u_i = 2 / sf.beta(λ_u_i, K_u_i+1)
    A_d_i = 1 / sf.beta(λ_d_i, K_d_i+1)
    I_u_i = A_u * sf.beta(λ_u_i+1, K_u_i+1)
    I_d_i = A_d * sf.beta(λ_d_i+1, K_d_i+1)
    u_V_i = Beta(λ_u_i, K_u_i+1)
    d_V_i = Beta(λ_d_i, K_d_i+1)

    remaining_i = 1 - (I_u_i + I_d_i)
    dirichlet_i = Dirichlet([3., 2., 1, 0.5, 0.3, 0.2, 0.1])
    I_i = rand(dirichlet_i) * remaining_i

    λ_g1_i = rand(Uniform(-1, 0))
    λ_g2_i = rand(Uniform(0, 1))
    K_g_i = rand(Uniform(2, 10))
    A_g2_i = I_i[1] / sf.beta(λ_g2_i+1, K_g_i+1)
    A_g1_i = I_i[2] / sf.beta(λ_g1_i+1, 5+1)

    λ_q_i = rand(Uniform(-1, 0))
    A_ubar_i = I_i[3] / (2 * sf.beta(λ_q_i+1, 5+1))
    A_dbar_i = I_i[4] / (2 * sf.beta(λ_q_i+1, 5+1))
    A_s_i = I_i[5] / (2 * sf.beta(λ_q_i+1, 5+1))
    A_c_i = I_i[6] / (2 * sf.beta(λ_q_i+1, 5+1))
    A_b_i = I_i[7] / (2 * sf.beta(λ_q_i+1, 5+1))

    total[i] = A_u_i * sf.beta(λ_u_i+1, K_u_i+1) + A_d_i * sf.beta(λ_d_i+1, K_d_i+1)
    total[i] += A_g1_i * sf.beta(λ_g1_i+1, 5+1) + A_g2_i * sf.beta(λ_g2_i+1, K_g_i+1)
    total[i] += 2 * (A_ubar_i + A_dbar_i + A_s_i + A_c_i + A_b_i) * sf.beta(λ_q_i+1, 5+1)

    xg2_i = A_g2_i * x.^λ_g2_i .* (1 .- x).^K_g_i
    xg1_i = A_g1_i * x.^λ_g1_i .* (1 .- x).^5
    plot!(x, [x .* A_u_i .* x.^λ_u_i .* (1 .- x).^K_u_i * 2], alpha=alpha, color="blue", lw=3)
    plot!(x, x .* A_d_i .* x.^λ_d_i .* (1 .- x).^K_d_i, alpha=alpha, color="orange", lw=3)
    plot!(x, xg1_i + xg2_i, alpha=alpha, color="green", lw=3)
    plot!(x, A_ubar_i * x.^λ_q_i .* (1 .- x).^5, alpha=alpha, color="red", lw=3)
    plot!(x, A_dbar_i * x.^λ_q_i .* (1 .- x).^5, alpha=alpha, color="purple", lw=3)
    plot!(x, A_s_i * x.^λ_q_i .* (1 .- x).^5, alpha=alpha, color="brown", lw=3)
    plot!(x, A_c_i * x.^λ_q_i .* (1 .- x).^5, alpha=alpha, color="pink", lw=3)
    plot!(x, A_b_i * x.^λ_q_i .* (1 .- x).^5, alpha=alpha, color="grey", lw=3)
end

plot!(xlabel="x", ylabel="x f(x)", xscale=:log, legend=false,
      ylims=(1e-8, 10), yscale=:log)

Looks like naive priors need some work...

PDF Parametrisation interface

PartonDensity provides a handy interface to the "Valence shape + Dirichlet" style parametrisation, as demonstrated here.

using PartonDensity

hyper_params = PDFParameters(λ_u=0.5, K_u=4.0, λ_d=0.6, K_d=6.0, λ_g1=-0.37, λ_g2=-0.7,
                             K_g=6.0, λ_q=0.5, seed=5, weights=[50., 0.5, 5., 5., 3., 2., 1.]);

plot_input_pdfs(hyper_params)

int_xtotx(hyper_params) ≈ 1

true

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